%I #7 Feb 13 2014 13:25:21
%S 1,2,3,4,8,5,10,22,14,6,28,62,38,16,7,78,174,106,44,18,9,220,492,298,
%T 124,50,24,11,622,1390,842,350,140,66,30,12,1758,3930,2380,988,394,
%U 186,84,32,13,4972,11114,6730,2794,1114,526,236,90,36,15,14062,31434
%N Dispersion of (2*floor(n*sqrt(2))), by antidiagonals.
%C Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n)), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence. Examples:
%C (1) s=A000040 (the primes), D=A114537, u=A114538.
%C (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C More recent examples of dispersions: A191426-A191455.
%e Northwest corner:
%e 1...2....4....10...28
%e 3...8....22...62...174
%e 5...14...38...106..298
%e 6...16...44...124..350
%e 7...18...50...140..394
%t (* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
%t r=40; r1=12; c=40; c1=12; f[n_] :=2*Floor[n*Sqrt[2]] (* complement of column 1 *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t (* A191541 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191541 sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y Cf. A114537, A035513, A035506, A191540.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 07 2011